Question about Euclid's parallel postulate (5th postulate).

In summary, the conversation is discussing the inability to prove Euclid's fifth postulate, which states that there is only one line that can be drawn parallel to a given line through a single point. The proposed proof of this postulate is found to be circular, as it relies on assuming the parallel postulate to prove itself. The conversation also touches on the concept of small scales and how the parallel postulate can be true in those instances, but may not hold true on larger scales or in non-Euclidean geometries. The validity and verification of the Pythagorean theorem is also debated, with one party arguing that it is always true in small scales and measurable, while the other party disagrees and states that it is not always true in
  • #1
parshyaa
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Why can't we prove euclids fifth postulate
What's wrong in this proof:
IMG_20170930_211402928.jpg

why can't we prove that there is only one line which passes through a single point which is parallel to a line.

If we can prove that two lines are parallel by proving that the alternate angles of a transverse passing through parallel side must be equal.
IMG_20170930_213900940.jpg

Then we could show that when a line throught that point behaves same then it will be parallel.

I know that euclids 5th postulate is unproveable and thus we have non-euclidean geometry, but i want to know what's wrong in my proof.

 
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  • #2
The external angle theorem, as you have stated it, is only provable if we assume the parallel postulate to be true. So your proposed proof of the parallel postulate is circular.
 
  • #3
phyzguy said:
The external angle theorem, as you have stated it, is only provable if we assume the parallel postulate to be true. So your proposed proof of the parallel postulate is circular.
How?
Tell me Where did i used parallel postulate to prove exterior angle theorem.
IMG_20171001_081657430.jpg
 
  • #5
Svein said:
When you say that the sum of the angles in a triangle is 180°.
So what do u want to say?
Let just stick to the euclidian geometry,
Then in euclidian geometry we can prove sum of angles of a triangle equals 180°
Because total angle of a straight line equals 180°.
What is wrong in my proof
So what do you want to justify, can't we prove total angle in a triangle equals 180° in euclidian geometry.
 
  • #6
Svein said:
When you say that the sum of the angles in a triangle is 180°.

In elliptic geometry (https://en.wikipedia.org/wiki/Elliptic_geometry) the sum of the angles is >180° and in hyperbolic geometry (https://en.wikipedia.org/wiki/Hyperbolic_geometry) the sum of the angles is <180°.
I think you are right
Because in proving parallel postulate i used angle sum property of triangle and in proving triamgle sum property i have to use parallel postulate and thus its absured
 
  • #7
When you say "Let's just stick to Euclidean Geometry", this is equivalent to assuming the Parallel Postulate. Euclid assumed the Parallel Postulate to be true. That's why it's called "Euclidean Geometry"
 
  • #8
Please note that in small scales Euclid's parallel postulate is always true.
It's only when great distances come into play that it's not true.
Otherwise we couldn't do maths and physics!
 
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  • #9
a-nobody said:
Please note that in small scales Euclid's parallel postulate is always true.
It's only when great distances come into play that it's not true.
Otherwise we couldn't do maths and physics!
What does small scale means to you
Does it changes in a plane for larger distances?
or it changes when distances are measured on earth(because Earth is spherical and value changes).
 
  • #10
a-nobody said:
Please note that in small scales Euclid's parallel postulate is always true.
That depends on how you phrase the parallel postulate.

"Given a line and a point not on that line there is exactly one line parallel to the given line containing the given point"?

Always false on small scales.
 
  • #11
Yes, always true. Why not? It gives rise to the Pythagorean Theorem always true and measurable and verifiable locally, in small everyday scales.
 
  • #12
a-nobody said:
Yes, always true. Why not? It gives rise to the Pythagorean Theorem always true and measurable and verifiable locally, in small everyday scales.
In mathematics, "true" and "approximately true" are not synonymous.
 
  • #13
jbriggs444 said:
In mathematics, "true" and "approximately true" are not synonymous.
In the context of what you said:

"Given a line and a point not on that line there is exactly one line parallel to the given line containing the given point"?

"Always false on small scales."

I've never encountered the Pythagorean Theorem to be true in the neighborhood of a point and false or approximately true in another.
 
  • #14
a-nobody said:
In the context of what you said:

"Given a line and a point not on that line there is exactly one line parallel to the given line containing the given point"?

"Always false on small scales."

I've never encountered the Pythagorean Theorem to be true in the neighborhood of a point and false or approximately true in another.
Two lines (in a plane) are "parallel" if they contain no points in common. Given a line and a point not on that line, there are many lines that are "parallel" to the given line and that contain the given point as long as we consider only the local area.

That has nothing to do with the Pythagorean theorem.

Now let us proceed to your claim that the Pythagorean theorem is true and verifiable as long as we restrict our attention to a local area. Consider, for example, a spherical geometry. For any non-degenerate triangle in the geometry, no matter how small, the Pythagorean theorem will always be false. It will never be verifiable by physical measurement.
 
  • #15
I respectfully disagree. Ever since Gauss and his magnificent "Disquisitiones generales circa superficies curvas" the area of the infinitesimal orthogonal triangle on the surface to find the curvature is taken to be: base x height / 2 completely on good terms with the Pythagorean Theorem.
I won't even mention that every manifold is homeomorphic to Euclidean space near a point.
By every physical measurement then as long as we probe close enough in the vicinity of a point, the Pythagorean Theorem is true. I hope that we are all in agreement that if the Pythagorean Theorem is true then the space is Euclidean and that there is a unique line parallel from some point to a nearby neighboring line. If there is a pencil of parallel lines to it, this becomes visible only as we approach infinity and the space deviates from Euclidean, if at all.
I really do not understand what your argument is about.
 
  • #16
a-nobody said:
I respectfully disagree. Ever since Gauss and his magnificent "Disquisitiones generales circa superficies curvas" the area of the infinitesimal orthogonal triangle on the surface to find the curvature is taken to be: base x height / 2 completely on good terms with the Pythagorean Theorem.
I won't even mention that every manifold is homeomorphic to Euclidean space near a point.
By every physical measurement then as long as we probe close enough in the vicinity of a point, the Pythagorean Theorem is true. I hope that we are all in agreement that if the Pythagorean Theorem is true then the space is Euclidean and that there is a unique line parallel from some point to a nearby neighboring line. If there is a pencil of parallel lines to it, this becomes visible only as we approach infinity and the space deviates from Euclidean, if at all.
I really do not understand what your argument is about.
The Pythagorean Theorem is false in spherical geometries. You cannot make it true by using smaller triangles. You can only make it "almost true".
 
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  • #17
phyzguy said:
The external angle theorem, as you have stated it, is only provable if we assume the parallel postulate to be true. So your proposed proof of the parallel postulate is circular.
... answers the original question.
phyzguy said:
When you say "Let's just stick to Euclidean Geometry", this is equivalent to assuming the Parallel Postulate. Euclid assumed the Parallel Postulate to be true. That's why it's called "Euclid
... is the only valid definition here.
jbriggs444 said:
Two lines (in a plane) are "parallel" if they contain no points in common. Given a line and a point not on that line, there are many lines that are "parallel" to the given line and that contain the given point as long as we consider only the local area.

That has nothing to do with the Pythagorean theorem.
... is correct and nothing has to be added.
a-nobody said:
I respectfully disagree.
This is your personal right. Fortunately it doesn't change the truth value of what has been said.
Ever since Gauss and his magnificent "Disquisitiones generales circa superficies curvas" the area of the infinitesimal orthogonal triangle on the surface to find the curvature is taken to be: base x height / 2 completely on good terms with the Pythagorean Theorem.
... So? Yes, Gauß worked as a land surveyor. Yes, nobody stated anything else. Yes, this involved non Euclidean geometry. However, non Euclidean geometry is not subject to this thread. Also the term local is completely inappropriate in this context. It seems that you are talking about some kind of manifolds, in which case some of your assertions might make sense. Unfortunately we deal with a manifold here, which is not only locally Euclidean, but globally. Furthermore we are neither interested in the topology, which is quite boring in a metric Euclidean space, nor in the analysis on it. Our space is globally flat and isomorphic, homeomorphic, diffeomorphic or whatever you like to ##\mathbb{R}^2##, because it is ##\mathbb{R}^2##.

Since the OP's question has been answered and this debate started to create confusion rather than clearance, I close it.
In the case someone of the participants wants to start a discussion on non Euclidean geometries on Riemannian manifolds, please state all given facts at the beginning of a new thread. Here we have a simple Euclidean plane and classic geometry. No add-ons needed.
 
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What is Euclid's parallel postulate?

Euclid's parallel postulate, also known as the 5th postulate, states that if a line intersects two other lines and the interior angles on the same side are less than two right angles, then the two lines will eventually intersect on that side.

Why is Euclid's parallel postulate important?

Euclid's parallel postulate is important because it is a fundamental concept in geometry and is used to prove many other theorems and propositions. It also helped pave the way for the development of non-Euclidean geometries.

What is an example of Euclid's parallel postulate in action?

An example of Euclid's parallel postulate is the fact that parallel lines on a flat surface, such as a sheet of paper, will never intersect, no matter how far they are extended.

Is Euclid's parallel postulate true?

Euclid's parallel postulate is a postulate, which means it is assumed to be true in order to prove other theorems. However, it has been debated and challenged by mathematicians throughout history and has led to the development of non-Euclidean geometries.

Why did Euclid include the parallel postulate in his Elements?

Euclid included the parallel postulate in his Elements because it was a commonly held belief at the time and he needed it as a starting point to prove other theorems. It was not until centuries later that mathematicians began to question its validity.

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